Elsevier

NeuroImage

Volume 28, Issue 4, December 2005, Pages 869-880
NeuroImage

Quantitative evaluation of three cortical surface flattening methods

https://doi.org/10.1016/j.neuroimage.2005.06.055Get rights and content

Abstract

During the past decade, several computational approaches have been proposed for the task of mapping highly convoluted surfaces of the human brain to simpler geometric objects such as a sphere or a topological disc. We report the results of a quantitative comparison of FreeSurfer, CirclePack, and LSCM with respect to measurements of geometric distortion and computational speed. Our results indicate that FreeSurfer performs best with respect to a global measurement of metric distortion, whereas LSCM performs best with respect to angular distortion and best in all but one case with a local measurement of metric distortion. FreeSurfer provides more homogeneous distribution of metric distortion across the whole cortex than CirclePack and LSCM. LSCM is the most computationally efficient algorithm for generating spherical maps, while CirclePack is extremely fast for generating planar maps from patches.

Introduction

Since the highly convoluted cerebral and cerebellar cortices are topologically equivalent to a two-dimensional sheet (topological sphere or disc), surface representations of the cortex should facilitate the visualization and analysis of functional activation data by preserving important geometrical and topological relationships. Moreover, surface representations which can be parameterized using two-dimensional coordinate systems (i.e., flat maps) may be useful for anatomically driven inter-subject registration (Van Essen et al., 1998).

Various methods have been proposed to inflate and/or flatten cortical brain surfaces: CARET (Drury et al., 1996), FreeSurfer (Fischl et al., 1999), the Laplace–Beltrami operator (Angenent et al., 1999), circle packing (via the CirclePack software) (Hurdal et al., 1999), harmonic energy minimization (Gu and Yau, 2002), and Least Squares Conformal Mapping (LSCM) (Ju et al., 2004). Although flattening a cortical surface necessarily introduces metric distortion due to the non-constant Gaussian curvature of the surface, it is possible to preserve local angular information (“conformality”) (Ahfors, 1996). We call a mapping that keeps conformality of the surfaces a “conformal mapping”. Pioneering work on numerical implementation of spherical conformal mapping was done by Brechbuhler et al. (1995) and Szekely et al. (1996) for the purpose of surface parameterization.

In order to quantify angular and metric distortion using conformal mapping techniques and non-conformal methods, we examined the performance of three published, freely available surface-mapping algorithms: FreeSurfer, CirclePack, and LSCM. All three methods can flatten user-defined patches and produce two-dimensional spherical surface maps of cortical hemispheres. CARET and FreeSurfer are similar in that both algorithms explicitly minimize metric distortion by solving a large-scale nonlinear optimization problem. The other four algorithms listed above produce discrete quasi-conformal maps. The Laplace–Beltrami operator, harmonic energy minimization, and LSCM are based on different but equivalent definitions of conformal mapping. They use both vertex connectivity and metric information, whereas tangency-based circle packing makes use of metric information from only the surface boundary, and otherwise depends only on the vertex connectivity of the surface mesh. All of the algorithms mentioned above work with the input surface meshes directly, and do not attempt to either reduce or subdivide the mesh.

Section snippets

Flattening techniques

Let K be a simply connected triangulated cortical surface {{vi}i=1n,T={Ti=(vi1,vi2,vi3)}i=1m} where {vi}i=1n is a set of n vertices with n ≥ 3 and T is a set of m triangles consisting of triples of vertices. Assume that K is consistently oriented. Then each triangle of T has a uniquely defined normal.

Let U represent the flattening function. Assume that U is linear on each triangle Ti, implying that the flattening can be uniquely determined by the mapping of the vertices of K. Let TiU=U(Ti)

Results and discussions

All three methods were run on a PC Linux workstation (1.67 GHz AMD Athlon XP CPU, 1.0 GB main memory) at the University of Minnesota. The default settings of the software were used. For the metric distortion computations, the sub-mesh associated with each vertex in the original mesh consisted of approximately 900 vertices with K = 15.

Conclusions

LSCM preserved local angular information (shape) during flattening whereas CirclePack did not perform as well as expected due to the fact that the triangles of the cortical meshes were not equilateral. Adjustments to the default (tangency) circle packing approach such as non-tangency circle packings which preserve hyperbolic inversive distance (Bowers and Hurdal, 2003) may improve these results; inversive distance can be computed as a function of metric data from the triangle mesh. For all of

Acknowledgments

This work is supported in part by NIH grant MH57180 and NSF grant DMS101339. Special thanks go to Dr. Bruce Fischl of Harvard University for assistance with FreeSurfer. The authors would also like to acknowledge Bill Wood from the Department of Mathematics at Florida State University for assistance with processing some of the data. The authors wish to thank the referees for their very helpful comments and suggestions.

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    Current address: Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

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